question_answer

The free end of a thread wound on a bobbin is passed round a nail A hammered into the wall. The thread is pulled at a constant velocity. Assuming pure rolling of bobbin, find the velocity \[{{v}_{0}}\] of the centre of the bobbin at the instant when the thread forms an angle a with the vertical.

A) \[\frac{vR}{R\sin \alpha -r}\]

B) \[\frac{vR}{R\sin \alpha +r}\]

C) \[\frac{2vR}{R\sin \alpha +r}\]

D) \[\frac{v}{R\sin \alpha +r}\]

Correct Answer: A

Solution :

[a] When the thread is pulled, the bobbin rolls to the right. Resultant velocity of point B along the thread is\[v={{v}_{0}}\sin \alpha -\omega r\], where\[{{v}_{0}}\sin \alpha \] is the component of translational velocity along the thread and \[\omega r\]linear velocity due to rotation. As the bobbin rolls without slipping, \[{{v}_{0}}=\omega R\]. Solving the obtained equations, we get\[{{v}_{0}}=\frac{vR}{R\,\sin \,\alpha -r}\]